The aim of the present work is to show the content of Plato’s unwritten “late teaching”
which constitutes the last stage in the evolutionary process of the development of his thought, the stage called “the unwritten (or esoteric) teaching” . It grew quite naturally from the earlier stages of Plato’s philosophy (“the written teaching”). Plato decided to focus his attention
on the ontological status of mathematics, with particular emphasis on the necessary conditions
for the existence of mathematics, and on the question about the obligatoriness, independent
from the subjective statuting, of mathematical theorems. As a result of this analysis,
“the theory of ideas”, formulated in the exoteric stage, became transformed into “the
theory of ideal numbers”. This is indeed the essence of the transformation that took place
in the late period of Platonic thinking, when the theory of ideas was interpreted in a mathematical
context, and served the purpose of explaining the ontological status of mathematical
objects. This situation resulted, first of all, from the role that mathematics played in
the Academy, where - as we happen to know - mathematical studies were an important
part of education, and where the most outstanding mathematicians of the time used to work.
The above is witnessed by the relation of Aristotle, a member of the Academy, who in his
Metaphysics contained some information about the philosophical disputes taking place there.
Let it be noticed that Aristotle’s criticism of Plato’s ideas refers in the first place to the
dispute on the status of mathematical objects, the special case of which we may find in the
M and N books (it seems that it is exactly for this reason that we cannot fully and properly
understand Aristotle’s attitude towards Plato’s theory of ideas if we fail to take into account
the matters connected with the ontology of mathematics).
In the late stage of Plato’s activity, we find a particularly strong confirmation of the
links between ontology and mathematics when analysing the documents connected with
Plato’s successors in the Old Academy: Speusippus, Xenocrates, Eudoxus, or Philip of Opus.
It can be seen very clearly to what extent they continue Plato’s “late thought” in becoming
involved in the dispute concerning the status of ontological principles, on the relations
between mathematical objects and the concept of mathematical natural sciences. A completion
of this picture we find in the documents of the “intermediate tradition” through which the ancient commentators of Plato tell us about the late form of his philosophy. We have here
again to do with the problem of analysing the links that exist between ontology, mathematics
and what can be called mathematical natural sciences. The said documents have been
collected by K. Gaiser under the general title Testimonia Platonica.
Bearing in mind the above mentioned facts, it seems justifiable to try to reconstruct
Plato’s late teaching, and its interpretation in the context of the theory associated with period
o f the dialogues. In this sense the present work is a continuation and expansion of the
analyses carried out in the work The Theory of Ideas - The Evolution of Plato ’s Thought
(Katowice 1997 [the first edition], and 1999 [the second edition]) where I attempted to justify
the thesis that “the theory of ideas” is not a random and heterogeneous collection of statements
and philosophical intuitions scattered all over various dialogues, but rather an orderly
process of development in which several stages can be discerned. I assumed then that
Plato’s thought has an evolutionary nature, and this evolution can be fully exemplified by
“the theory of ideas” . This is because Plato started to construct his theory on the basis of
Socratic inspirations, completing Socrates’ conception with an ontological dimension. “The
theory of ideas” thus constructed in the middle-Academic period, was later subjected to
a thorough analysis and reinterpretation, and given a new form that was called “the theory
of ideal numbers”. Since the latter demanded its own, ultimate legitimisation, Plato decided
to adopt the conception called “the theory of ontological principles” . However, neither
“the theory of ideal numbers” nor “the theory of ontological principles” can be observed
in Plato’s dialogues. Plato’s disciples and ancient commentators described these theories
only later. This is why it became conventional to call this ultimate, late form of Plato’s
thought by the name of “the unwritten teaching”. Its essence was already in ancient times
an object of interpretative controversies. The modem research on Plato’s thought brought
about a great intensification of that controversy. Some scholars try to belittle the significance
of “the unwritten teaching” (among other names Cherniss, M. Insardi-Parente, G.
Reale, G. Vlastos, J.N. Findlay, or E. Dont), but others emphasise its importance for the
understanding of Plato’s entire philosophy (here belong H J. Krämer, G. Kaiser, T.A. Szlezal,
G. Reale, G. Halfwassen, J.N. Findlay, or V. Hosle). The representatives of the latter position
assumed that, in order to fully understand the philosophical sense of the dialogues,
it is necessary to refer to “the unwritten teaching”, so that it was accepted that the said
teaching contains the essence of the Platonic thought. In the present work, I suggest a
different interpretative stand. I assume, namely, that “the unwritten teaching” (the theory
o f ideal numbers and theory of principles) constitute a natural consequence of the development
o f Plato’s ontological thought, and should be considered from the point of view
o f that evolution, that is in the context of the maturation of his thought. Thus, “the unwritten
teaching” appears the crowning of Plato’s ontological conceptions. The point is that
already in the period of the dialogues we can see Plato grappling with the difficulties inherent
in his “theory of ideas”, which is confirmed by the disputes within the Academy, and debates
with representatives of other philosophical positions. Plato seeks new solutions, striving
to find the conclusive arguments, which results in new conceptions. Consequently, there
arose “the theory of ideal numbers” and “theory of principles” . How can we not recognise
that we have to do here with a peculiar case of the evolution of the philosopher’s views?
Do we find a philosopher whose thought would not undergo a process of this kind? How
can we then claim that, in the light of the testimony of the disciples and ancient commentators,
“the unwritten teaching” do not constitute an essential part of Plato’s doctrine? And
how can we, on the other hand, maintain that it is only “the unwritten teaching” that express
the Platonic thought? Particularly strange appears to be the statement that Plato, already
when he was writing his dialogues, possessed a conception proper to “the unwritten teaching”,
but decided not to reveal it, so that we should reconstruct it on the basis of an analysis of the dialogues. I think that the above-described positions, original as they may be, are
rather radical and thus not necessarily corresponding to the actual state of affairs. The suggestion
that emphasises the evolution of Plato’s views, and the development of his thought
stemming from his being aware of the limitations connected with particular aspects of his
theory, and his desire to legitimise it, may seem then more natural, even though it may be
less original. I shall try to justify this thesis in the present work, using as an example an
analysis of the links between the Platonic concepts of ontology and mathematics. Thus, the
present dissertation finds its place in the debate, conducted by the most outstanding modem
commentators, on our understanding of Plato’s philosophy. It would be useless to enumerate
here those commentators, their names and conceptions will turn up while discussing various
particular problems. I have made an effort to use their research achievements by integrating
them with my analyses, or by entering into a discussion with them. I have tried
to distance myself only from such interpretations that are based on much later theories (particularly
the contemporary ones) whose connection with Plato’s thought seems dubious.
I mean particularly such interpretations that stipulate the reading of Plato from the point
of view of modem logical, mathematical, and philosophical theories. It is impossible,
naturally, to become liberated from the temporal context in which a given text is commented
upon. We should, however, strive to limit such conditioning so that a given interpretation
is to a greater extent predicated on the inner logic of the original text than on the commentator’s
(however inevitable) assumptions. Consequently, the present analyses are based
on Plato’s own writings and on those of ancient commentators and doxographers informing
us of his conceptions.